Problem: The equation of hyperbola $H$ is $\dfrac {(x-9)^{2}}{16}-\dfrac {(y+1)^{2}}{81} = 1$. What are the asymptotes?
We want to rewrite the equation in terms of $y$ , so start off by moving the $y$ terms to one side: $\dfrac {(y+1)^{2}}{81} = - 1 + \dfrac {(x-9)^{2}}{16}$ Multiply both sides of the equation by $81$ $(y+1)^{2} = { - 81 + \dfrac{ (x-9)^{2} \cdot 81 }{16}}$ Take the square root of both sides. $\sqrt{(y+1)^{2}} = \pm \sqrt { - 81 + \dfrac{ (x-9)^{2} \cdot 81 }{16}}$ $ y + 1 = \pm \sqrt { - 81 + \dfrac{ (x-9)^{2} \cdot 81 }{16}}$ As $x$ approaches positive or negative infinity, the constant term in the square root matters less and less, so we can just ignore it. $y + 1 \approx \pm \sqrt {\dfrac{ (x-9)^{2} \cdot 81 }{16}}$ $y + 1 \approx \pm \left(\dfrac{9 \cdot (x - 9)}{4}\right)$ Subtract $1$ from both sides and rewrite as an equality in terms of $y$ to get the equation of the asymptotes: $y = \pm \dfrac{9}{4}(x - 9) -1$